L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.618 + 1.07i)3-s + (−0.499 − 0.866i)4-s + (−0.618 + 1.07i)5-s − 1.23·6-s + 0.999·8-s + (0.736 − 1.27i)9-s + (−0.618 − 1.07i)10-s + (−0.5 − 0.866i)11-s + (0.618 − 1.07i)12-s + 3.23·13-s − 1.52·15-s + (−0.5 + 0.866i)16-s + (1.23 + 2.14i)17-s + (0.736 + 1.27i)18-s + (−3.61 + 6.26i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.356 + 0.618i)3-s + (−0.249 − 0.433i)4-s + (−0.276 + 0.478i)5-s − 0.504·6-s + 0.353·8-s + (0.245 − 0.424i)9-s + (−0.195 − 0.338i)10-s + (−0.150 − 0.261i)11-s + (0.178 − 0.309i)12-s + 0.897·13-s − 0.394·15-s + (−0.125 + 0.216i)16-s + (0.299 + 0.519i)17-s + (0.173 + 0.300i)18-s + (−0.830 + 1.43i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.313552497\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.313552497\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
good | 3 | \( 1 + (-0.618 - 1.07i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (0.618 - 1.07i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 + (-1.23 - 2.14i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.61 - 6.26i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.47 - 6.01i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2.47T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + (1 - 1.73i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.23 + 7.33i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.38 - 2.39i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.381 - 0.661i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.70 + 9.88i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 + (-6.47 - 11.2i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + (-5 + 8.66i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.18824214213550617391147731837, −9.318406927062400887380992912564, −8.390999269856611857305806699835, −7.986279241967213139257046611562, −6.71524132039146494495187675943, −6.20289312483282726747417484916, −5.07127180662542370636488856294, −3.87276448199071481435993295039, −3.32176166096398450224484290843, −1.47464238243376668959017661907,
0.68207113210597681583455588215, 1.98511233931925031107968086101, 2.91373245706495228419903590258, 4.26636334225929755746766076681, 4.96190310199200306036564025225, 6.44051708591140015663688149942, 7.21649986973800667243716720091, 8.242728125109100708815554622752, 8.553874080778250762494923888649, 9.505949188266758020401587695183